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Full text of "Mathematical And Physical Papers - Iii"

222             ON THE CONDUCTION  OF  HEAT  IN  CRYSTALS.
drical surfaces whose generating lines are parallel to the diameter of the thermic ellipsoid which is conjugate to the plane of the plate.
17. The state of temperature, under given circumstances, of a rectangular parallelepiped formed of an uncrystallized substance, may be determined by certain known formulas which it is not necessary here to describe.
Hence, the state of temperature of a parallelepiped cut from a crystal in such a manner that its edges are parallel to a system of conjugate diameters of the thermic ellipsoid may be determined by the same formula*. This parallelepiped will of course be oblique-angled, except in the particular case in which its edges are parallel to the thermic axes. It may be remarked that a parallelepiped for which the state of temperature shall be deter-minable by the formulas in question may be cut from a crystal in a manner quite as general as from an uncrystallized substance. In both cases the direction of the first edge is arbitrary, and, when it is fixed on, the plane of the other two edges is determined in direction. The direction of the second edge having been chosen arbitrarily in the plane above mentioned, that of the third edge is determined.
It does not seem worth while to notice the crystalline figures derived from spheres, &c., on account of the mechanical difficulty attending their execution. Besides, the derivation presents no theoretical difficulty.
Further consideration of the general expressions for the flux.
18. It has been already remarked, that if the crystal possess two planes of symmetry, the nine arbitrary constants which appear in the expressions for the flux in three rectangular directions, from which the flux in any other direction may be derived, reduce themselves to six, and the expressions for the flux take the form (8). I proceed now to consider what grounds we have for believing that these expressions, with only six arbitrary constants, are the most general possible.
In the first place, it may be observed that this result follows